Stefanie Gerke, Gregory Gutin, Anders Yeo, Yacong Zhou
{"title":"Lower Bounds for Maximum Weight Bisections of Weighted Triangle-Free Subcubic Graphs","authors":"Stefanie Gerke, Gregory Gutin, Anders Yeo, Yacong Zhou","doi":"10.1002/jgt.70056","DOIUrl":null,"url":null,"abstract":"<p>A bisection of a graph is a cut in which the number of vertices in the two parts of the cut differ by at most 1. In this paper, we consider maximum weight bisections of edge-weighted triangle-free subcubic graphs and show that every weighted triangle-free subcubic graph <span></span><math>\n <semantics>\n <mrow>\n \n <mi>G</mi>\n \n <mo>=</mo>\n \n <mrow>\n <mo>(</mo>\n \n <mrow>\n <mi>V</mi>\n \n <mo>,</mo>\n \n <mi>E</mi>\n \n <mo>,</mo>\n \n <mi>w</mi>\n </mrow>\n \n <mo>)</mo>\n </mrow>\n </mrow>\n <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt70056:jgt70056-math-0001\" wiley:location=\"equation/jgt70056-math-0001.png\"><mrow><mi>G</mi><mo>=</mo><mrow><mo>(</mo><mrow><mi>V</mi><mo>,</mo><mi>E</mi><mo>,</mo><mi>w</mi></mrow><mo>)</mo></mrow></mrow></math></annotation>\n </semantics></math> has a bisection with weight at least <span></span><math>\n <semantics>\n <mrow>\n \n <mi>θ</mi>\n \n <mo>⋅</mo>\n \n <mi>w</mi>\n \n <mrow>\n <mo>(</mo>\n \n <mi>G</mi>\n \n <mo>)</mo>\n </mrow>\n </mrow>\n <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt70056:jgt70056-math-0002\" wiley:location=\"equation/jgt70056-math-0002.png\"><mrow><mi>\\unicode{x003B8}</mi><mo>\\unicode{x022C5}</mo><mi>w</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></annotation>\n </semantics></math> unless <span></span><math>\n <semantics>\n <mrow>\n \n <mi>G</mi>\n \n <mo>≅</mo>\n \n <msub>\n <mi>K</mi>\n \n <mrow>\n <mn>1</mn>\n \n <mo>,</mo>\n \n <mn>3</mn>\n </mrow>\n </msub>\n </mrow>\n <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt70056:jgt70056-math-0003\" wiley:location=\"equation/jgt70056-math-0003.png\"><mrow><mi>G</mi><mo>\\unicode{x02245}</mo><msub><mi>K</mi><mrow><mn>1</mn><mo>,</mo><mn>3</mn></mrow></msub></mrow></math></annotation>\n </semantics></math> (where <span></span><math>\n <semantics>\n <mrow>\n \n <mi>θ</mi>\n \n <mo>=</mo>\n \n <mfrac>\n <mn>613</mn>\n \n <mn>855</mn>\n </mfrac>\n \n <mo>≈</mo>\n \n <mn>0.716959</mn>\n </mrow>\n <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt70056:jgt70056-math-0004\" wiley:location=\"equation/jgt70056-math-0004.png\"><mrow><mi>\\unicode{x003B8}</mi><mo>=</mo><mfrac><mn>613</mn><mn>855</mn></mfrac><mo>\\unicode{x02248}</mo><mn>0.716959</mn></mrow></math></annotation>\n </semantics></math>). We conjecture that <span></span><math>\n <semantics>\n <mrow>\n \n <mi>θ</mi>\n </mrow>\n <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt70056:jgt70056-math-0005\" wiley:location=\"equation/jgt70056-math-0005.png\"><mrow><mi>\\unicode{x003B8}</mi></mrow></math></annotation>\n </semantics></math> can be replaced by <span></span><math>\n <semantics>\n <mrow>\n \n <mn>11</mn>\n \n <mo>/</mo>\n \n <mn>15</mn>\n \n <mo>≈</mo>\n \n <mn>0.73333</mn>\n </mrow>\n <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt70056:jgt70056-math-0006\" wiley:location=\"equation/jgt70056-math-0006.png\"><mrow><mn>11</mn><mo>/</mo><mn>15</mn><mo>\\unicode{x02248}</mo><mn>0.73333</mn></mrow></math></annotation>\n </semantics></math>, the value of <span></span><math>\n <semantics>\n <mrow>\n \n <mi>θ</mi>\n </mrow>\n <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt70056:jgt70056-math-0007\" wiley:location=\"equation/jgt70056-math-0007.png\"><mrow><mi>\\unicode{x003B8}</mi></mrow></math></annotation>\n </semantics></math> for the Peterson graph and prove the conjecture for weighted bridgeless triangle-free cubic graphs.</p>","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"113 2","pages":"252-273"},"PeriodicalIF":1.0000,"publicationDate":"2026-08-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/jgt.70056","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Graph Theory","FirstCategoryId":"100","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1002/jgt.70056","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"2026/5/14 0:00:00","PubModel":"Epub","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
A bisection of a graph is a cut in which the number of vertices in the two parts of the cut differ by at most 1. In this paper, we consider maximum weight bisections of edge-weighted triangle-free subcubic graphs and show that every weighted triangle-free subcubic graph has a bisection with weight at least unless (where ). We conjecture that can be replaced by , the value of for the Peterson graph and prove the conjecture for weighted bridgeless triangle-free cubic graphs.
期刊介绍：
The Journal of Graph Theory is devoted to a variety of topics in graph theory, such as structural results about graphs, graph algorithms with theoretical emphasis, and discrete optimization on graphs. The scope of the journal also includes related areas in combinatorics and the interaction of graph theory with other mathematical sciences.
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